


Of Temperament and Irrationality

by DesireeArmfeldt



Category: Arcadia - Stoppard
Genre: Documentation, F/M, Mathematics, Music, Pre-Slash
Language: English
Status: Completed
Published: 2020-05-01
Updated: 2020-05-01
Packaged: 2021-03-01 19:22:03
Rating: Teen And Up Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 1,290
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/23862229
Author URL: https://archiveofourown.org/users/DesireeArmfeldt/pseuds/DesireeArmfeldt
Summary: Thomasina and Septimus explore Pythagorean tuning and equal temperament.
Relationships: Lady Croom/Septimus Hodge (implied), Thomasina Coverly/Septimus Hodge
Comments: 4
Kudos: 5
Collections: Wayback Exchange 2020





	Of Temperament and Irrationality

**Author's Note:**

  * For [innie](https://archiveofourown.org/users/innie/gifts).



> In canon, Thomasina says she’s 13 years and 10 months old on April 10th 1809. Therefore her birthday is in August, and she is 15 years and 4 months in December 1810.

10 December, 1810

An Essay Upon the Theme, “The Pythagorean Method of Tuning” by Thomasina Coverly

Pythagoras, we are told, laid forth the geometrical principles of music, following the principle that the ear is best pleased by a pair of notes whose frequencies are related by a ratio of two integers, the smaller the integers, the more pleasing. An octave represents a ratio of two to one, and thus is most pleasing, while a fifth represents a ratio of three to two, which is nearly as good, and allows the construction of scales from which melodies and harmonies can be formed.

One may generate all the twelve notes that make up an octave by this method: start from the root frequency, C for example, multiply by 3/2 to get the frequency a fifth above, then multiply the result to produce a fifth above that fifth, and so on, until one arrives back at the root of the key in a much higher octave.

From this point on, I shall omit the arithmetical calculations of the exact figures, for is not the entire purpose of algebra to spare us the tedium of arithmetic to as great an extent as we can manage to wring from it?

> **_Your tutor admits to having said as much in a moment of unconsidered candor, and it is the pupil’s prerogative to take license where it is given. However, all license has its limits and it is the wise pupil who divines those limits and avoids the consequences of surpassing them._ **

To get all of the notes together into the span of an octave, we must transpose most of them downward by one or more octaves. This is accomplished by multiplying the frequency of a particular pitch by ½, that is, the ratio of the original frequency to that of the same note, an octave lower. The first and second notes, differing only by a fifth, are already in the same octave; the same is true of the next pair, and so forth. Each successive pair, together, must be lowered by an additional octave, thus:

And so on, until you reach C again, 6 octaves up from where you started:

The C major scale, by the by, consists of the notes C, D, E, F, G, A, B and C again, and may be generated by the method of adding fifths just described, except that one must start the sequence with F to make it come out right. Why Pythagoras or anyone else decided that these seven notes, and not the other five, comprise a scale, I do not know. I suppose the reason is recorded in the history of music, unless it has been lost like the hundreds of Athenian plays we know were once committed to paper, and the thousands upon thousands of things we are not even aware that once someone knew.

However, the main difficulty with the Pythagorean system is the fact that the frequency of this higher C should, by definition, be greater than that of the original by a factor of 2 to the 6th power, which is not the case. For 2 cannot go evenly into 3, no matter how many of each you have multiplied together beforehand. They remain relatively prime and therefore, as unwilling to consort as gentlemen and ladies at a ball who have no one to introduce them.

As Archimedes could not square the circle exactly, but could calculate an approximation to it, so Pythagoras threw up his hands and settled on cramming his scale into an octave by breaking his own rules, giving scant measure on the final interval so as to keep the ratio of C to C at 2 to 1. This short interval was called the “wolf tone” for its dissonant sound and, for the same reason, was avoided in composition. Like a stair with a broken step, Pythagoras’s scales could be used to create music, so long as the musician minded where he trod and avoided falling into the pit of dissonance.

The difficulty does not lie in the mathematics, but in the determination of Pythagoras to do the impossible. God made the laws of nature and of mathematics and constructed the human ear, but ‘twas mankind decided to prefer simple ratios of frequencies to more complex relationships between pitches, like Romeo and Juliet setting their hearts on the one person they were forbidden to love.

Eventually, people got tired of hopping around the hole and invented another way to generate scales, in which all the treads are even and one may start and stop at any note one pleases without mishap. And now, nobody minds that the intervals between keys on the pianoforte are all the twelfth root of 2, so long as they may listen to the music of Herr Bach and all the rest. Indeed, most of us cannot tell when the instrument falls out of tune, to the great annoyance of those few who can, like my Aunt Sophia, or Jellaby, who, of course, would never say such a thing, but I have seen him wince when he passes the music room door, and it is never long after such occasions that the piano tuner is sent for.

The moral of this lesson is that man cannot make 3/2 divide evenly into 2 by wishing it so, and that one who sets his heart on the impossible must either bow to reason or make himself a cheat and a liar, practicing deceit upon himself before all others, singing the praises of beauty while sticking his fingers in his ears to ignore the howling of the wolf tone.

> **_A clear and succinct explanation of the mathematics we discussed._ ** **_We shall continue plumbing the depths of the circle of fifths tomorrow, which may shed light upon the place of F in the C major scale._ ** ****
> 
> **_High marks as well for giving your essay a coherent metaphor and moral. However, consider that many have lamented the loss of the modes with their distinct colors of tone and mood, that resulted from the ancient, uneven temperament of musical instruments. The wolf tone, perhaps, served as a reminder that in this mortal world, there is no pleasure without pain, and that death lurks even in Arcadia. Erase it from the scale, put it from our minds as best we may, yet there is no more escaping this truth of nature than the indivisibility of 3 by 2._ **
> 
> **_Furthermore, consider that for Pythagoras, numbers such as the twelfth root of two, which could not be expressed as ratios of two integers, were a meaningless impossibility—literally, irrational. It is embracing the irrational that has made possible the precision of equal-tempered tuning, as well as the manufacture of machinery such as the steam engine our Mr. Noakes dreams of unleashing upon your father’s grounds, the better to realize his vision of Nature, wild, untamed, and ungovernable. If there be beauty in his manufactured wilderness, does it spring from the predictability of the machine or the irrationality of the fractions that make it possible? From Mr. Noakes’ rational calculations or from the wild fancies of Mr. Walpole and Mrs. Radcliffe?_ **
> 
> **_No doubt Romeo would have been better off had he gone home to bed instead of attending the ball, got up and went about his business in the morning never knowing what he had missed, and married some nameless heiress, while Juliet was wed to Paris, bore him seven children, and died a fat and contented woman of fourscore. But it is doubtful that audiences would have flocked to see the play, had Shakespeare written it thus._ **
> 
> **_The heart yearns after beauty, and cannot be satisfied on a diet of pure reason._ **


End file.
